Numerical Analysis –MTH603 VU
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Jacobi’s Method
Definition
An n x n matrix [A] is said to be orthogonal if
1
[][] [],
i.e.[] []
T
T
A
AI
A
A−
=
=
In order to compute all the eigen values and the corresponding eigenvectors of a real symmetric matrix,
Jacobi’s method is highly recommended. It is based on an important property from matrix theory, which
states
that, if [A] is an n x n real symmetric matrix, its eigen values are real, and there exists an orthogonal matrix
[S] such that the diagonal matrix D is
1
[][][]SAS
−
This digitalization can be carried out by applying a series of orthogonal transformations
12
, ,..., ,
n
SS S
Let A be an n x n real symmetric matrix. Suppose ij
abe numerically the largest element amongst the off-
diagonal elements of A. We construct an orthogonal matrix S1 defined as
sin , sin ,
cos , cos
ij ji
ii jj
ss
ss
θ
θ
θ
θ
=−=
==
While each of the remaining off-diagonal elements are zero, the remaining diagonal elements are assumed
to be unity. Thus, we construct S1 as under
1
i-th column -th column
10 0 0 0
01 0 0 0
0 0 cos sin 0 i-th row
0 0 sin cos 0 -th row
00 0 0 1
j
S
j
θθ
θθ
↓↓
−←
=
←
"" "
"" "
## # # #
"" "
## # # #
"" "
## # # #
"" "
Where cos , sin , sin cosand
θ
θθ ϑ
− are inserted in ( , ),( , ),( , ), ( , ) thii i j ji j j
−
positions respectively,
and elsewhere it is identical with a unit matrix.
Now, we compute
1
11111
T
DSASSAS
−
==
Since S1 is an orthogonal matrix, such that .After the transformation, the elements at the position (i , j), (j ,
i) get annihilated, that is dij and dji reduce to zero, which is seen as follows:
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